
Since we assume that the functions \({\color {black}\mathbb {E}_{\xi _i^k}}[F(x_i,{\color {black}\xi _i^k})] + f_0(x_i)\)  are strongly convex and have Lipschitz continuous gradients (cf. Assumption REF  and REF ), by [1]} we have

\(& -2\langle \nabla {\color {black}\mathbb {E}_{\xi _i^k}}[F(x_i^k,\xi _i^k)] + \nabla f_0(x_i^k) - \nabla {\color {black}\mathbb {E}_{\xi _i^k}}[F(x_i^*,\xi _i^k)] - \nabla f_0(x_i^*), x_i^k-x_i^*\rangle \\\le & -\frac{2u_iL_i}{u_i+L_i} \Vert x_i^k-x_i^*\Vert ^2 -\frac{2}{u_i+L_i}\Vert \nabla {\color {black}\mathbb {E}_{\xi _i^k}}[F(x_i^k,\xi _i^k)] + \nabla f_0(x_i^K) - \nabla {\color {black}\mathbb {E}_{\xi _i^k}}[F(x_i^*,\xi _i^k)]^2 - \nabla f_0(x_i^*) \Vert ^2. \nonumber \) 
